That is the Riemann-Lebesgue Lemma. You can find the proof on page 103 of "Papa Rudin" (Real and Complex Analysis). You can also go here for an outline of the proof.
Any help with the following is really appreciated (hints, as well!):
For and , let
Prove that as
We're allowed to use the 'strong operator convergence' theorem for Banach spaces, as well as other standard results in Banach spaces.